Title: Kernel Smoothing Tools for Philology and Historical Dialectology
Version: 0.2
URL: http://www.icge.co.uk/
Description: Contains kernel smoothing tools designed for use by historical dialectologists and philologists for exploring spatial and temporal patterns in noisy historical language data, such as that obtained from historical texts. The main way in which these might differ from other implementations of kernel smoothing is that they assume that the function (linguistic variable) being explored has the form of the relative frequency of a series of discrete possibilities (linguistic variants). This package also offers a way of exploring distributions in 2-dimensional space and in time with separate kernels, and tools for identifying appropriate bandwidths for these.
License: GPL (≥ 3)
Encoding: UTF-8
Imports: benchmarkme, directlabels, dplyr, ggplot2, grDevices, gridExtra, Hmisc, pbapply, reshape2, rlang, stats, terra, utils, wordspace
RoxygenNote: 7.3.3
NeedsCompilation: no
Packaged: 2025-11-09 17:37:11 UTC; tamsinn
Author: Tamsin Blaxter ORCID iD [aut, cre]
Maintainer: Tamsin Blaxter <tamsinblaxter@gmail.com>
Repository: CRAN
Date/Publication: 2025-11-12 21:10:02 UTC

Calculate temporal bandwidths by spatial resolution

Description

This function calculates relationships between temporal bandwidth and possible spatial resolution for a given power and suggests minimum possible temporal bandwidth for a given resolution

Usage

calculate.bandwidths.by.resolution(
  dataset,
  dependent.variable = "dependent.variable",
  time = "year",
  weight = "weight",
  alpha = 0.05,
  margin = 0.1,
  measure.times,
  temporal.bandwidth.limits,
  temporal.bandwidth.n.levels = 200,
  minimum.spatial.resolution = 5,
  summary.plots = FALSE,
  kernel.function = gaussian.kernel
)

Arguments

dataset

The dataset to be smoothed as a data.frame.

dependent.variable

String name of the column in dataset with the dependent variable (defaults to "dependent.variable"); this column should be numeric or factor.

time

String name of the column in dataset with the time variable (defaults to "year").

weight

String name of column in the dataset with numeric weights (defaults to "weight").

alpha

Numeric alpha for calculating error margins (defaults to 0.05).

margin

Numeric desired error margin for calculating spatial bandwidths (defaults to 0.1).

measure.times

A numeric vector of specific times at which to make estimates; if not given, will default to seq(from=min(time),to=max(time),length.out=5).

temporal.bandwidth.limits

Numeric vector of length 2 specifying minimum and maximum temporal bandwidth to be tested (defaults to the range of time0.01 to the range of time2).

temporal.bandwidth.n.levels

Number of distinct levels of temporal bandwidth to be tested (defaults to 200).

minimum.spatial.resolution

Numeric minimum spatial resolution.

summary.plots

If TRUE, plots of smoothed sample density, the dependent variable, and variance are returned along with the plot of resolution by bandwidth.

kernel.function

The kernel function, one of gaussian.kernel, gaussian.square.kernel, triangular.kernel, square.kernel, or a custom function (defaults to gaussian.kernel).

Value

A list containing a plot of spatial resolution by temporal bandwidth, along with other summary plots of the data if summary.plots==TRUE, and the calculated minimum temporal bandwidth.

Calculate temporal bandwidths by separated points

Description

This function calculates relationships between temporal bandwidth and spatial bandwidths at a series of specified points for a given power and suggests minimum possible temporal bandwidth such that bandwidths at those points are never greater than 2.2365the distance to the nearest point (for gaussian kernels) or 2that distance (for other kernels)

Usage

calculate.bandwidths.by.separated.points(
  dataset,
  dependent.variable = "dependent.variable",
  x = "x",
  y = "y",
  time = "year",
  weight = "weight",
  alpha = 0.05,
  margin = 0.1,
  separated.points,
  measure.times,
  temporal.bandwidth.limits,
  temporal.bandwidth.n.levels = 200,
  kernel.function = gaussian.kernel,
  projection = NA,
  include.visualisation = FALSE,
  separated.points.labels,
  round.up.low.variance = FALSE
)

Arguments

dataset

The dataset to be smoothed as a data.frame.

dependent.variable

String name of the column in dataset with the dependent variable (defaults to "dependent.variable"); this column should be numeric or factor.

x

String name of column containing numeric x co-ordinate (defaults to "x").

y

String name of column containing numeric y co-ordinate (defaults to "y").

time

String name of the column in dataset with the time variable (defaults to "year").

weight

String name of column in the dataset with numeric weights (defaults to "weight").

alpha

Numeric alpha for calculating error margins (defaults to 0.05).

margin

Numeric desired error margin for calculating spatial bandwidths (defaults to 0.1).

separated.points

Data.frame containing two columns same names as x,y in dataset with x and y coordinates of points to be kept separate.

measure.times

A numeric vector of specific times at which to make estimates; if not given, will default to seq(from=min(time),to=max(time),length.out=5).

temporal.bandwidth.limits

A numeric vector of length 2 specifying minimum and maximum temporal bandwidth to be tested (defaults to the range of time0.01 to the range of time2).

temporal.bandwidth.n.levels

Number of distinct levels of temporal bandwidth to be tested (defaults to 200).

kernel.function

The kernel function, one of gaussian.kernel, gaussian.square.kernel, triangular.kernel, square.kernel, or a custom function (defaults to gaussian.kernel).

projection

A spatial projection as a proj4 string - if given, data will be projected before smoothing and results will be deprojected before returning.

include.visualisation

If TRUE, will return a ggplot visualisation.

separated.points.labels

String vector of the names of the separated points (used in the visualisation).

round.up.low.variance

Set to TRUE if there are periods of time with extremely low variance.

Value

A list with suggested bandwidth and the choke point and time, plus a visualisation of bandwidths and resolutions if include.visualisation==TRUE.

Gaussian kernel

Description

Gaussian kernel function - weights all points by distance on a Gaussian curve with standard deviation of bandwidth.

Usage

gaussian.kernel(d, bandwidth)

Arguments

d

Numeric vector of distances.

bandwidth

Numeric bandwidth.

Value

A numeric vector of kernel weights.

Gaussian square kernel

Description

Gaussian square kernel function - weights all points within bandwidth by distance on a Gaussian curve with standard deviation of bandwidth, but all points beyond bandwidth at 0.

Usage

gaussian.square.kernel(d, bandwidth)

Arguments

d

Numeric vector of distances.

bandwidth

Numeric bandwidth.

Value

A numeric vector of kernel weights.

Kernel smooth data in space alone

Description

This function performs kernel smoothing on a dataset in space alone.

Usage

kernel.smooth.in.space(
  dataset,
  dependent.variable = "dependent.variable",
  x = "x",
  y = "y",
  weight = "weight",
  normalise.by,
  data.type = "factor",
  alpha = 0.05,
  margin = 0.1,
  kernel.function = gaussian.kernel,
  adaptive.spatial.bw = TRUE,
  measure.points,
  projection = NA,
  round.up.low.variance = TRUE,
  explicit = TRUE
)

Arguments

dataset

Dataset to be smoothed as a data.frame.

dependent.variable

String name of the single column in dataset with the factor dependent variable (if data.type=="factor") or a vector of column names with numeric counts (if data.type=="count") (defaults to "dependent.variable").

x

String name of column containing numeric x co-ordinate (defaults to "x").

y

String name of column containing numeric y co-ordinate (defaults to "y").

weight

String name of column in the dataset with numeric weights (defaults to "weight").

normalise.by

String name of column by which data should be normalised (typically factor with document, speaker or writer ids).

data.type

The type of the dependent variable: either "factor", if each row is a token, or "count", if each row is a document, speaker or writer with token counts in separate columns (defaults to "factor").

alpha

Numeric alpha for calculating error margins (defaults to 0.05).

margin

Numeric desired error margin for calculating spatial bandwidths (defaults to 0.1).

kernel.function

The kernel function, one of gaussian.kernel, gaussian.square.kernel, triangular.kernel, square.kernel, or a custom function (defaults to gaussian.kernel).

adaptive.spatial.bw

A boolean indicating whether the spatial bandwidth is adaptive (set to achieve margin at every point) or static (set to the average of bandwidths needed to achieve margin at every point).

measure.points

A data.frame of spatial points at which estimates are to be made, with two columns with the same names as x,y in dataset; if not supplied, estimates are at the same locations as dataset.

projection

The spatial projection as a proj4 string - if given, data will be projected before smoothing and results will be deprojected before returning.

round.up.low.variance

Set to TRUE if there are periods of time with extremely low variance (defaults to TRUE).

explicit

If TRUE, progress will be reported with a progress bar (defaults to TRUE).

Value

A data.frame with the smoothed estimates.

Examples

n=400;
synthesised.data<-data.frame(x=stats::runif(n),y=stats::runif(n),
    year=stats::runif(n,0,sqrt(2)));
synthesised.data$dependent.variable<-unlist(lapply(1:nrow(synthesised.data),
    function(X){
    stats::dist(as.matrix(synthesised.data[c(1,X),1:2]),method =
        "euclidean")<synthesised.data$year[X];
}))
result<-kernelPhil::kernel.smooth.in.space(dataset = synthesised.data);
ggplot2::ggplot(result,ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point();

Kernel smooth data in space and time

Description

This function performs kernel smoothing on a dataset in time and space. A static temporal kernel is applied first, and then an (optionally) adaptive spatial kernel on this weighted data.

Usage

kernel.smooth.in.space.and.time(
  dataset,
  dependent.variable = "dependent.variable",
  x = "x",
  y = "y",
  time = "year",
  weight = "weight",
  normalise.by,
  data.type = "factor",
  alpha = 0.05,
  margin = 0.1,
  kernel.function = gaussian.kernel,
  adaptive.spatial.bw = TRUE,
  temporal.bandwidth,
  measure.points,
  measure.times,
  projection = NA,
  explicit = TRUE
)

Arguments

dataset

The dataset to be smoothed as a data.frame.

dependent.variable

String name of the single column in dataset with the factor dependent variable (if data.type=="factor") or a vector of column names with numeric counts (if data.type=="count") (defaults to "dependent.variable").

x

String name of column containing numeric x co-ordinate (defaults to "x").

y

String name of column containing numeric y co-ordinate (defaults to "y").

time

String name of the column in dataset with the time variable (defaults to "year").

weight

String name of column in the dataset with numeric weights (defaults to "weight").

normalise.by

String name of column by which data should be normalised (typically factor with document, speaker or writer ids).

data.type

The type of the dependent variable as a string: either "factor", if each row is a token, or "count", if each row is a document, speaker or writer with token counts in separate columns (defaults to "factor").

alpha

Numeric alpha for calculating error margins (defaults to 0.05).

margin

Numeric desired error margin for calculating spatial bandwidths (defaults to 0.1).

kernel.function

The kernel function, one of gaussian.kernel, gaussian.square.kernel, triangular.kernel, square.kernel, or a custom function (defaults to gaussian.kernel).

adaptive.spatial.bw

Boolean indicating whether the spatial bandwidth is adaptive (set to achieve margin at every point) or static (set to the average of bandwidths needed to achieve margin at every point).

temporal.bandwidth

Numeric bandwidth of the (gaussian) temporal kernel.

measure.points

A data.frame of spatial points at which estimates are to be made, with two columns with the same names as x,y in dataset; if not supplied, estimates are at the same locations as dataset.

measure.times

A numeric vector of specific times at which to make estimates; if not given, will default to seq(from=min(time),to=max(time),length.out=5).

projection

Spatial projection as a proj4 string - if given, data will be projected before smoothing and results will be deprojected before returning.

explicit

If TRUE, progress will be reported with a progress bar (defaults to TRUE).

Value

A list containing the parameters and a data.frame with the smoothed estimates.

Examples

n=200;
synthesised.data<-data.frame(x=stats::runif(n),y=stats::runif(n),
    year=stats::runif(n,0,sqrt(2)));
synthesised.data$dependent.variable<-unlist(lapply(1:nrow(synthesised.data),
    function(X){
    stats::dist(as.matrix(synthesised.data[c(1,X),1:2]),method =
        "euclidean")<synthesised.data$year[X];
}));
result<-kernelPhil::kernel.smooth.in.space.and.time(dataset =
    synthesised.data,temporal.bandwidth = 0.25,measure.times =
    seq(from=-0.05,to=1.15,length.out=4),alpha = 0.15,margin = 0.2);
gridExtra::grid.arrange(ggplot2::ggplot(result$results[[1]],
    ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point(),ggplot2::ggplot(result$results[[2]],
    ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point(),ggplot2::ggplot(result$results[[3]],
    ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point(),ggplot2::ggplot(result$results[[4]],
    ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point());

Kernel smooth data in space and time, returning specific error margins at each point

Description

This function performs kernel smoothing on a dataset in time and space. A static temporal kernel is applied first, and then an (optionally) adaptive spatial kernel on this weighted data. Note that this is the same as kernel.smooth.in.space.and.time() except that it returns specific error margins with every estimate and is much slower.

Usage

kernel.smooth.in.space.and.time.with.margins(
  dataset,
  dependent.variable = "dependent.variable",
  x = "x",
  y = "y",
  time = "year",
  weight = "weight",
  normalise.by,
  data.type = "factor",
  alpha = 0.05,
  margin = 0.1,
  kernel.function = gaussian.kernel,
  adaptive.spatial.bw = TRUE,
  temporal.bandwidth,
  measure.points,
  measure.times,
  projection = NA,
  explicit = TRUE
)

Arguments

dataset

The dataset to be smoothed as a data.frame.

dependent.variable

String name of the single column in dataset with the factor dependent variable (if data.type=="factor") or a vector of column names with numeric counts (if data.type=="count") (defaults to "dependent.variable").

x

String name of column containing numeric x co-ordinate (defaults to "x").

y

String name of column containing numeric y co-ordinate (defaults to "y").

time

String name of the column in dataset with the time variable (defaults to "year").

weight

String name of column in the dataset with numeric weights (defaults to "weight").

normalise.by

String name of column by which data should be normalised (typically factor with document, speaker or writer ids).

data.type

The type of the dependent variable as a string: either "factor", if each row is a token, or "count", if each row is a document, speaker or writer with token counts in separate columns (defaults to "factor").

alpha

Numeric alpha for calculating error margins (defaults to 0.05).

margin

Numeric desired error margin for calculating spatial bandwidths.

kernel.function

The kernel function, one of gaussian.kernel, gaussian.square.kernel, triangular.kernel, square.kernel, or a custom function (defaults to gaussian.kernel).

adaptive.spatial.bw

Boolean indicating whether the spatial bandwidth is adaptive (set to achieve margin at every point) or static (set to the average of bandwidths needed to achieve margin at every point).

temporal.bandwidth

Numeric bandwidth of the (gaussian) temporal kernel.

measure.points

A data.frame of spatial points at which estimates are to be made, with two columns with the same names as x,y in dataset; if not supplied, estimates are at the same locations as dataset.

measure.times

A numeric vector of specific times at which to make estimates; if not given, will default to seq(from=min(time),to=max(time),length.out=5).

projection

The spatial projection as a proj4 string - if given, data will be projected before smoothing and results will be deprojected before returning.

explicit

If TRUE, progress will be reported with a progress bar (defaults to TRUE).

Value

A list containing the parameters and a data.frame with the smoothed estimates.

Examples

n=200;
synthesised.data<-data.frame(x=stats::runif(n),y=stats::runif(n),
    year=stats::runif(n,0,sqrt(2)));
synthesised.data$dependent.variable<-unlist(lapply(1:nrow(synthesised.data),
    function(X){
    stats::dist(as.matrix(synthesised.data[c(1,X),1:2]),method =
        "euclidean")<synthesised.data$year[X];
}));
result<-kernelPhil::kernel.smooth.in.space.and.time.with.margins(dataset =
    synthesised.data,temporal.bandwidth = 0.2,measure.times =
    seq(from=0.15,to=0.85,length.out=2),alpha=0.4,margin=0.2);
gridExtra::grid.arrange(ggplot2::ggplot(result$results[[1]],
    ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point(),ggplot2::ggplot(result$results[[2]],
    ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point())

Kernel smooth data in space alone, returning specific error margins at each point

Description

This function performs kernel smoothing on a dataset in space alone. It is the same as kernel.smooth.in.space(), except that the results include the error margins for the estimates at every point. Note that it is much slower than kernel.smooth.in.space().

Usage

kernel.smooth.in.space.with.margins(
  dataset,
  dependent.variable = "dependent.variable",
  x = "x",
  y = "y",
  weight = "weight",
  normalise.by,
  data.type = "factor",
  alpha = 0.05,
  margin = 0.1,
  kernel.function = gaussian.kernel,
  adaptive.spatial.bw = TRUE,
  measure.points,
  projection = NA,
  round.up.low.variance = TRUE,
  explicit = TRUE
)

Arguments

dataset

The dataset to be smoothed as a data.frame.

dependent.variable

String name of the single column in dataset with the factor dependent variable (if data.type=="factor") or a vector of column names with numeric counts (if data.type=="count") (defaults to "dependent.variable").

x

String name of column containing numeric x co-ordinate (defaults to "x").

y

String name of column containing numeric y co-ordinate (defaults to "y").

weight

String name of column in the dataset with numeric weights (defaults to "weight").

normalise.by

String name of column by which data should be normalised (typically factor with document, speaker or writer ids).

data.type

The type of the dependent variable as a string: either "factor", if each row is a token, or "count", if each row is a document, speaker or writer with token counts in separate columns (defaults to "factor").

alpha

Numeric alpha for calculating error margins (defaults to 0.05).

margin

Numeric desired error margin for calculating spatial bandwidths (defaults to 0.1).

kernel.function

The kernel function, one of gaussian.kernel, gaussian.square.kernel, triangular.kernel, square.kernel, or a custom function (defaults to gaussian.kernel).

adaptive.spatial.bw

A boolean indicating whether the spatial bandwidth is adaptive (set to achieve margin at every point) or static (set to the average of bandwidths needed to achieve margin at every point).

measure.points

A data.frame of spatial points at which estimates are to be made, with two columns with the same names as x,y in dataset; if not supplied, estimates are at the same locations as dataset.

projection

The spatial projection as a proj4 string - if given, data will be projected before smoothing and results will be deprojected before returning.

round.up.low.variance

Set to TRUE if there are periods of time with extremely low variance (defaults to TRUE).

explicit

If TRUE, progress will be reported with a progress bar (defaults to TRUE).

Value

A data.frame with the smoothed estimates.

Examples

n=400;
synthesised.data<-data.frame(x=stats::runif(n),y=stats::runif(n),
    year=stats::runif(n,0,sqrt(2)));
synthesised.data$dependent.variable<-unlist(lapply(1:nrow(synthesised.data),
    function(X){
    stats::dist(as.matrix(synthesised.data[c(1,X),1:2]),method =
        "euclidean")<synthesised.data$year[X];
}))
result<-kernelPhil::kernel.smooth.in.space.with.margins(dataset = synthesised.data);
ggplot2::ggplot(result,ggplot2::aes(x=x,y=y,colour=relative_density_TRUE))+
    ggplot2::geom_point();

Kernel smooth data in time alone

Description

This function performs kernel smoothing on a dataset in time alone.

Usage

kernel.smooth.in.time(
  dataset,
  dependent.variable = "dependent.variable",
  time = "year",
  weight = "weight",
  bandwidth = 10,
  sample.density.threshold = 3,
  length.out = 1000,
  alpha = 0.05,
  xlabel = "year",
  ylabel,
  greyscale = "compatible",
  save.path = "",
  measure.times,
  kernel.function = gaussian.kernel
)

Arguments

dataset

The dataset to be smoothed as a data.frame.

dependent.variable

String name of the column in dataset with the dependent variable (defaults to "dependent.variable"); this column should be numeric or factor.

time

String name of the column in dataset with the time variable (defaults to "year").

weight

String name of column in the dataset with numeric weights (defaults to "weight").

bandwidth

Numeric bandwidth of the kernel function.

sample.density.threshold

Numeric local density of samples below which no estimates will be returned.

length.out

The number of measure points along the time axis (defaults to 1000).

alpha

Numeric alpha for calculating error margins (defaults to 0.05).

xlabel

String label for the x-axis in returned plot (defaults to "year").

ylabel

String label for the y-axis in returned plot.

greyscale

If TRUE, plot will be in greyscale; if "compatible", plot will use a colour spectrum which also goes light>dark; otherwise, will use a non-greyscale-compatible colour scale.

save.path

String path to save plot to (if not given, plot will not be saved).

measure.times

A numeric vector of specific times at which to make estimates; if given, sample.density.threshold and length.out will be ignored.

kernel.function

The kernel function, one of gaussian.kernel, gaussian.square.kernel, triangular.kernel, square.kernel, or a custom function.

Value

A list containing a data.frame with the smoothed estimates, and a ggplot grob visualising them.

Examples

n=1000;
synthesised.data<-data.frame(x=stats::runif(n),y=stats::runif(n),
    year=stats::runif(n,0,sqrt(2)));
synthesised.data$dependent.variable<-unlist(lapply(1:nrow(synthesised.data),
    function(X){
    stats::dist(as.matrix(synthesised.data[c(1,X),1:2]),method =
    "euclidean")<synthesised.data$year[X];
}))
result<-kernelPhil::kernel.smooth.in.time(dataset = synthesised.data,
    bandwidth = 0.05,sample.density.threshold = 100);
result$plot;

Load kernel smooths

Description

Loads the output of kernel.smooth.in.space.and.time() or kernel.smooth.in.space.and.time.with.margins() previously saved with save.kernel.smooths()

Usage

load.kernel.smooths(location)

Arguments

location

String location to which results were saved.

Value

A list containing the parameters and a data.frame with the smoothed estimates (same structure as returned by kernel.smooth.in.space.and.time() and kernel.smooth.in.space.and.time.with.margins()).

Identify nearest point in time to a given estimate value

Description

This function takes the output of kernel.smooth.in.time and identifies the point in time when the smoothed estimate comes closest to some specific value. This is useful for tasks like identifying the likely midpoint of a change.

Usage

nearest.point(kernel.smooths, density, variant, n = 1, timerange)

Arguments

kernel.smooths

A list output by kernel.smooth.in.time().

density

The value of the dependent variable for which a time is to be identified.

variant

If the dependent variable was a factor, which level is being examined (do not give a value if dependent variable was numeric).

n

The number of nearest points to be returned (useful if the estimates cross the relevant threshold multiple times, defauls to 1).

timerange

Numeric vector of length two - used to restrict search to a specific time range within the kernel.smooth.in.time(), in the form c(min,max).

Value

One or more numeric values.

Sample size

Description

Returns necessary sample size for a given power and dataset

Usage

sample.size(alpha, margin, dependent.variable, weights)

Arguments

alpha

Numeric alpha.

margin

Numeric error margin.

dependent.variable

The variable being sampled as a vector.

weights

Numeric weights (if present) as a vector.

Value

A list containing the parameters and a data.frame with the smoothed estimates.

Sample size floored variance

Description

Returns necessary sample size for a given power and dataset, but treats any standard deviation below 0.1 as 0.1

Usage

sample.size.floored.variance(alpha, margin, dependent.variable, weights)

Arguments

alpha

Numeric alpha.

margin

Numeric error margin.

dependent.variable

The variable being sampled as a vector.

weights

Numeric weights (if present) as a vector.

Value

A list containing the parameters and a data.frame with the smoothed estimates.

Save kernel smooths in space and time

Description

Saves the output of kernel.smooth.in.space.and.time() or kernel.smooth.in.space.and.time.with.margins() to a directory

Usage

save.kernel.smooths(kernel.smooth, location, variable.name)

Arguments

kernel.smooth

A list output of kernel.smooth.in.space.and.time() or kernel.smooth.in.space.and.time.with.margins().

location

String location on the disk to save output to.

variable.name

String name of the variable (used in filenames).

Value

No returned value, called to save data to disk.

Square kernel

Description

Square kernel function - weights all points within bandwidth as 1, all others as 0.

Usage

square.kernel(d, bandwidth)

Arguments

d

Numeric vector of distances.

bandwidth

Numeric bandwidth.

Value

A numeric vector of kernel weights.

Triangular kernel

Description

Triangular kernel function - weights all points within bandwidth on a straight line such that where d=0 weight=1, and where d=bandwidth weight=0; all points beyond bandwidth are weighted 0.

Usage

triangular.kernel(d, bandwidth)

Arguments

d

Numeric vector of distances.

bandwidth

Numeric bandwidth.

Value

A numeric vector of kernel weights.